Representations of simple noncommutative Jordan superalgebras I

Abstract

In this article we begin the study of representations of simple finite-dimensional noncommutative Jordan superalgebras. In the case of degree $\geq 3$ we show that any finite-dimensional representation is completely reducible and, depending on the superalgebra, quasiassociative or Jordan. Then we study representations of superalgebras $D_t(\alpha,\beta,\gamma)$ and $K_3(\alpha, \beta, \gamma)$ and prove the Kronecker factorization theorem for superalgebras $D_t(\alpha,\beta,\gamma)$. In the last section we use a new approach to study noncommutative Jordan representations of simple Jordan superalgebras.